Properties

 Conductor 24 Order 2 Real Yes Primitive No Parity Odd Orbit Label 48.h

Related objects

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(48)
sage: chi = H[41]
pari: [g,chi] = znchar(Mod(41,48))

Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 24 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 2 Real = Yes sage: chi.is_primitive() pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = No sage: chi.is_odd() pari: zncharisodd(g,chi) Parity = Odd Orbit label = 48.h Orbit index = 8

Galois orbit

sage: chi.sage_character().galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

Inducingprimitive character

$$\chi_{24}(5,\cdot)$$ = $$\displaystyle\left(\frac{-24}{\bullet}\right)$$

Values on generators

$$(31,37,17)$$ → $$(1,-1,-1)$$

Values

 -1 1 5 7 11 13 17 19 23 25 29 31 $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$1$$
value at  e.g. 2

Related number fields

 Field of values $$\Q$$

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 48 }(41,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{48}(41,\cdot)) = \sum_{r\in \Z/48\Z} \chi_{48}(41,r) e\left(\frac{r}{24}\right) = 9.7979589711i$$

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 48 }(41,·),\chi_{ 48 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{48}(41,\cdot),\chi_{48}(1,\cdot)) = \sum_{r\in \Z/48\Z} \chi_{48}(41,r) \chi_{48}(1,1-r) = 0$$

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 48 }(41,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{48}(41,·)) = \sum_{r \in \Z/48\Z} \chi_{48}(41,r) e\left(\frac{1 r + 2 r^{-1}}{48}\right) = -0.0$$